Integrand size = 46, antiderivative size = 17 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=2+\frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \] Output:
2+(x+1/ln(5/4*x)^2)/ln(x)
Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \] Input:
Integrate[(-2*Log[x] - Log[(5*x)/4] + (-x + x*Log[x])*Log[(5*x)/4]^3)/(x*L og[x]^2*Log[(5*x)/4]^3),x]
Output:
(x + Log[(5*x)/4]^(-2))/Log[x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(x \log (x)-x) \log ^3\left (\frac {5 x}{4}\right )-\log \left (\frac {5 x}{4}\right )-2 \log (x)}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {2}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )}+\frac {\log (x)-1}{\log ^2(x)}-\frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )}dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )}dx+\frac {x}{\log (x)}\) |
Input:
Int[(-2*Log[x] - Log[(5*x)/4] + (-x + x*Log[x])*Log[(5*x)/4]^3)/(x*Log[x]^ 2*Log[(5*x)/4]^3),x]
Output:
$Aborted
Time = 1.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41
method | result | size |
parallelrisch | \(\frac {6+6 \ln \left (\frac {5 x}{4}\right )^{2} x}{6 \ln \left (\frac {5 x}{4}\right )^{2} \ln \left (x \right )}\) | \(24\) |
risch | \(\frac {4+16 x \ln \left (2\right )^{2}+4 x \ln \left (x \right )^{2}+4 x \ln \left (5\right )^{2}+8 x \ln \left (5\right ) \ln \left (x \right )-16 x \ln \left (2\right ) \ln \left (x \right )-16 \ln \left (5\right ) \ln \left (2\right ) x}{\left (2 \ln \left (5\right )-4 \ln \left (2\right )+2 \ln \left (x \right )\right )^{2} \ln \left (x \right )}\) | \(65\) |
parts | \(\text {Expression too large to display}\) | \(1911\) |
default | \(\text {Expression too large to display}\) | \(2586\) |
Input:
int(((x*ln(x)-x)*ln(5/4*x)^3-ln(5/4*x)-2*ln(x))/x/ln(x)^2/ln(5/4*x)^3,x,me thod=_RETURNVERBOSE)
Output:
1/6*(6+6*ln(5/4*x)^2*x)/ln(5/4*x)^2/ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x \log \left (\frac {5}{4}\right )^{2} + 2 \, x \log \left (\frac {5}{4}\right ) \log \left (x\right ) + x \log \left (x\right )^{2} + 1}{\log \left (\frac {5}{4}\right )^{2} \log \left (x\right ) + 2 \, \log \left (\frac {5}{4}\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}} \] Input:
integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5 /4*x)^3,x, algorithm="fricas")
Output:
(x*log(5/4)^2 + 2*x*log(5/4)*log(x) + x*log(x)^2 + 1)/(log(5/4)^2*log(x) + 2*log(5/4)*log(x)^2 + log(x)^3)
Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (15) = 30\).
Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 5.53 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x \log {\left (x \right )}^{2} - 4 x \log {\left (2 \right )} \log {\left (5 \right )} + 4 x \log {\left (2 \right )}^{2} + x \log {\left (5 \right )}^{2} + \left (- 4 x \log {\left (2 \right )} + 2 x \log {\left (5 \right )}\right ) \log {\left (x \right )} + 1}{\log {\left (x \right )}^{3} + \left (- 4 \log {\left (2 \right )} + 2 \log {\left (5 \right )}\right ) \log {\left (x \right )}^{2} + \left (- 4 \log {\left (2 \right )} \log {\left (5 \right )} + 4 \log {\left (2 \right )}^{2} + \log {\left (5 \right )}^{2}\right ) \log {\left (x \right )}} \] Input:
integrate(((-x+x*ln(x))*ln(5/4*x)**3-ln(5/4*x)-2*ln(x))/x/ln(x)**2/ln(5/4* x)**3,x)
Output:
(x*log(x)**2 - 4*x*log(2)*log(5) + 4*x*log(2)**2 + x*log(5)**2 + (-4*x*log (2) + 2*x*log(5))*log(x) + 1)/(log(x)**3 + (-4*log(2) + 2*log(5))*log(x)** 2 + (-4*log(2)*log(5) + 4*log(2)**2 + log(5)**2)*log(x))
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 186, normalized size of antiderivative = 10.94 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=-\frac {3 \, \log \left (5\right ) - 6 \, \log \left (2\right ) + 2 \, \log \left (x\right )}{\log \left (5\right )^{4} - 8 \, \log \left (5\right )^{3} \log \left (2\right ) + 24 \, \log \left (5\right )^{2} \log \left (2\right )^{2} - 32 \, \log \left (5\right ) \log \left (2\right )^{3} + 16 \, \log \left (2\right )^{4} + {\left (\log \left (5\right )^{2} - 4 \, \log \left (5\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (\log \left (5\right )^{3} - 6 \, \log \left (5\right )^{2} \log \left (2\right ) + 12 \, \log \left (5\right ) \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3}\right )} \log \left (x\right )} + \frac {\log \left (5\right ) - 2 \, \log \left (2\right ) + 2 \, \log \left (x\right )}{{\left (\log \left (5\right )^{2} - 4 \, \log \left (5\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} + {\left (\log \left (5\right )^{3} - 6 \, \log \left (5\right )^{2} \log \left (2\right ) + 12 \, \log \left (5\right ) \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3}\right )} \log \left (x\right )} + {\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right ) \] Input:
integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5 /4*x)^3,x, algorithm="maxima")
Output:
-(3*log(5) - 6*log(2) + 2*log(x))/(log(5)^4 - 8*log(5)^3*log(2) + 24*log(5 )^2*log(2)^2 - 32*log(5)*log(2)^3 + 16*log(2)^4 + (log(5)^2 - 4*log(5)*log (2) + 4*log(2)^2)*log(x)^2 + 2*(log(5)^3 - 6*log(5)^2*log(2) + 12*log(5)*l og(2)^2 - 8*log(2)^3)*log(x)) + (log(5) - 2*log(2) + 2*log(x))/((log(5)^2 - 4*log(5)*log(2) + 4*log(2)^2)*log(x)^2 + (log(5)^3 - 6*log(5)^2*log(2) + 12*log(5)*log(2)^2 - 8*log(2)^3)*log(x)) + Ei(log(x)) - gamma(-1, -log(x) )
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (15) = 30\).
Time = 0.14 (sec) , antiderivative size = 187, normalized size of antiderivative = 11.00 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x \log \left (5\right )^{2} - 4 \, x \log \left (5\right ) \log \left (2\right ) + 4 \, x \log \left (2\right )^{2} + 1}{2 \, \log \left (5\right )^{2} \log \left (2\right ) - 8 \, \log \left (5\right ) \log \left (2\right )^{2} + 8 \, \log \left (2\right )^{3} + \log \left (5\right )^{2} \log \left (\frac {1}{4} \, x\right ) - 4 \, \log \left (5\right ) \log \left (2\right ) \log \left (\frac {1}{4} \, x\right ) + 4 \, \log \left (2\right )^{2} \log \left (\frac {1}{4} \, x\right )} - \frac {2 \, \log \left (5\right ) - 2 \, \log \left (2\right ) + \log \left (\frac {1}{4} \, x\right )}{\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} \log \left (2\right ) + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 2 \, \log \left (5\right )^{3} \log \left (\frac {1}{4} \, x\right ) - 8 \, \log \left (5\right )^{2} \log \left (2\right ) \log \left (\frac {1}{4} \, x\right ) + 8 \, \log \left (5\right ) \log \left (2\right )^{2} \log \left (\frac {1}{4} \, x\right ) + \log \left (5\right )^{2} \log \left (\frac {1}{4} \, x\right )^{2} - 4 \, \log \left (5\right ) \log \left (2\right ) \log \left (\frac {1}{4} \, x\right )^{2} + 4 \, \log \left (2\right )^{2} \log \left (\frac {1}{4} \, x\right )^{2}} \] Input:
integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5 /4*x)^3,x, algorithm="giac")
Output:
(x*log(5)^2 - 4*x*log(5)*log(2) + 4*x*log(2)^2 + 1)/(2*log(5)^2*log(2) - 8 *log(5)*log(2)^2 + 8*log(2)^3 + log(5)^2*log(1/4*x) - 4*log(5)*log(2)*log( 1/4*x) + 4*log(2)^2*log(1/4*x)) - (2*log(5) - 2*log(2) + log(1/4*x))/(log( 5)^4 - 4*log(5)^3*log(2) + 4*log(5)^2*log(2)^2 + 2*log(5)^3*log(1/4*x) - 8 *log(5)^2*log(2)*log(1/4*x) + 8*log(5)*log(2)^2*log(1/4*x) + log(5)^2*log( 1/4*x)^2 - 4*log(5)*log(2)*log(1/4*x)^2 + 4*log(2)^2*log(1/4*x)^2)
Time = 1.58 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x\,{\ln \left (\frac {5\,x}{4}\right )}^2+1}{{\ln \left (\frac {5\,x}{4}\right )}^2\,\ln \left (x\right )} \] Input:
int(-(log((5*x)/4) + 2*log(x) + log((5*x)/4)^3*(x - x*log(x)))/(x*log((5*x )/4)^3*log(x)^2),x)
Output:
(x*log((5*x)/4)^2 + 1)/(log((5*x)/4)^2*log(x))
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {\mathrm {log}\left (\frac {5 x}{4}\right )^{2} x +1}{\mathrm {log}\left (\frac {5 x}{4}\right )^{2} \mathrm {log}\left (x \right )} \] Input:
int(((-x+x*log(x))*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x) ^3,x)
Output:
(log((5*x)/4)**2*x + 1)/(log((5*x)/4)**2*log(x))